摘要
本文主要围绕乘积微分算子的白伴性及特征值对边界的依赖性展开研究.
微分算子从本质来说是无界可闭的线性算子,无界闭的线性算子的定义域一定不能是全空间,因此定义域的选择始终是微分算子研究中的一个十分重要而困难的问题.在微分算式给定的前提下,对所研究的算子提出的具体要求最终体现在对定义域的限制上.定义域不同的微分算子,其谱分解,特别是离散谱会有很大不同.在这些定义域的选择中,自伴域的选择就是其中重要之一.自伴微分算子因其有重要的应用背景,不仅使得它的谱与反谱问题成为数学家研究的热门课题,同时白伴性的识别与描述问题也被提到了重要位置.
本文首先研究了微分算式乘积的自伴域的实参数解描述问题,在适当条件的假设下,利用互为相反数的一对值所对应的解刻画了微分算式乘积的自伴域,使得自伴边界条件中矩阵的确定只与这些解在正则点的初始值有关.
其次,对于四阶奇型对称微分算式而言,会出现中间亏指数情形.本文接着研究了由具有任意亏指数的对称常微分算式生成的两个四阶及高阶奇型微分算子的积的自伴性问题.通过在半直线上使用实参数解对自伴域的刻画定理及分析技巧,以矩阵形式给出了,具任意亏指数的奇型对称微分算式产生的两个微分算子的积自伴的充要条件,并获得了与积算子自伴性有关的一些结果.
再次,人们在工程实践中发现：一根材料均匀的,横截面积与长度相比可忽略不计的,有弹性的杆,两端以一定的有意义的方式固定住,然后去弹奏它,会发现杆发出的音会随其长度的缩短而逐渐变强,即杆的固有频率在逐渐增高,这一现象更为力学家所熟知.用数学的语言将这一问题翻译出来就是四阶边值问题的特征值对边界的依赖性问题.结合Dauge, Q. Kong ([38],[51],[87])等人的工作,借助微分算子的谱理论这一有利工具我们研究了两类四阶及高阶边值问题的特征值对边界的依赖性.给出了第n个特征值关于其中一端点的一阶微分表达式,并证明了当区间长度趋于零时,在本文所考虑的边界条件情形下,所有的特征值会趋于无穷.并给出了具体的例子.
最后本文研究了具有周期边界条件的四阶边值问题的矩阵表示,并考虑了它的逆过程即矩阵特征值问题的四阶边值问题表示.
全文共分六个部分：一、介绍本文所研究问题的背景及本文的主要结果；二、文中所涉及相关符号、概念以及性质；三、微分算式乘积的自伴域的实参数解刻画；四、两个奇型微分算子乘积的自伴性；五、微分方程边值问题的特征值对边界的依赖性；六、具有周期边界条件的四阶边值问题的矩阵表示.
In this paper, we study the self-adjointness of product of differential operators and the dependence of eigenvalues on the boundary. The dif-ferential operators are essentially the unbounded closable operators, and the domains of the unbounded closable operators must not be the whole space. Hence the choice of the domains of the differential operators are always important and difficult. Given a differential expression, the specific demands for the differential operators eventually reflect on the restrictions on the domains. Differential operators with different domains will have different spectral distributions, especially the discrete spectrum. Among the choice of the domains, the choice of self-adjoint domains is one of the important. Self-adjoint differential operator, because of its important ap-plication background, not only makes its spectrum with inverse spectrum problem become a hot topic of mathematicians, at the same time the prob-lem of identification and description of self-adjointness was also mentioned the important position.
Firstly we consider the problem of description of self-adjoint domains of product of differential expression in terms of real-parameter solutions. Under appropriate assumptions, using solutions corresponding to value of a pair of opposite each other, we present a characterization of self-adjoint boundary conditions for product of differential expression, making a matrix of self-adjoint boundary conditions be determined only associated with the initial value of the solutions in the regular point.
For fourth order singular symmetric differential expression, there will be a middle deficiency indices case. Next we study the problem of self-adjointness of product of two fourth-order and higher-order differential operators in middle deficiency indices case. By using the theorem of char-acterization of real parameter solutions on half line for self-adjoint domains, and analytical skills, we give a sufficient and necessary condition in the form of matrix, and get some results related to the self-adjointness of product operators.
Again, in the engineering practice people note:a elastic rod, its di-ameter are negligible compared with its length and both ends are fixed in some meaningful way, then we go to play it and find that the sound stem from the rod will gradually strengthen with its length shortening, that is, the natural frequency of the rod increases gradually, this phenomenon is highly known by dynamicists. Using mathematical language, we translate this problem to the problem of dependence of eigenvalues of fourth-order boundary value problems on the boundary. With the help of spectrum theory of differential operator, combining the work of Dauge, Q., Kong and others ([38],[51],[87]), we study the dependence of eigenvalues of two kinds of fourth order boundary value problems and a kind of higher order boundary value problems on the boundary. We give specific form of the equations for derivative of the nth eigenvalue as a function of an endpoint, and proved that as the length of the interval shrinks to zero all eigenvalues march off to plus infinity for all boundary conditions considered in this paper. In addition, we give some examples.
Finally, we study the matrix representations of fourth order boundary value problems with periodic boundary conditions, and consider its inverse process, i.e. the representations of fourth-order boundary value problems of matrix eigenvalue problem.
This paper contains six parts.1. The background and main results in this paper;2. The associated fundamental definitions and important lemmas;3. Characterization of real-parameter solutions of self-adjoint do-mains for the product of differential expressions;4. The self-adjointness of product of two singular differential operators;5. The dependence of eigen-values of differential equation boundary value problems on the boundary;6. Matrix representations of fourth order boundary value problems with periodic boundary conditions.
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